X-ray computed tomography apparatus

ABSTRACT

An x-ray computed tomography apparatus is operated so that a reconstruction of arbitrarily selectable volume regions can be accomplished. A Fourier reconstruction is implemented based on parallel data in planes that are inclined by the angle φ relative to a plane perpendicular to the z axis.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention is directed to an x-ray computed tomographyapparatus of the type having an annular x-ray source surrounding ameasuring field, the annular x-ray source having an annular anode thatis scanned by an electron beam for generating a rotating x-ray beam. Thegeometry of such an x-ray computed tomography apparatus shall bereferred to below as EBT (electron beam tomography) geometry.

2. Description of the Prior Art

An especially fast scanning of an examination subject is possible withan x-ray computed tomography apparatus of the above type, so that motionunsharpness is largely suppressed. In order for the x-ray beam to enterunimpeded into the measuring field wherein the examination subject lies,the radiation detector that is likewise annularly fashioned and iscomposed of a row of detector elements arranged laterally next to theexit window of the x-ray beam. This permits that the x-ray beam toemerge unimpeded from this window and to be incident on the x-raydetector after leaving the measuring field. To this end, the x-ray beamis inclined at an angle relative to its rotational axis that deviatesslightly from 90°.

SUMMARY OF THE INVENTION

It is an object of the present invention to provide an x-ray computedtomography apparatus of the type generally described above wherein afast, artifact-free image reconstruction is achieved.

The above object is achieved in accordance with the principles of thepresent invention in an x-ray computed tomography apparatus of the typegenerally described above which produces, for each scan, a set ofmeasured values f(u_(i),p_(k),_(l)) for each scan for each projectionangle _(l) and each position u^(i) =z_(i) cos φ (wherein φ is the anglewhich the projection plane makes relative to the x-y plane of aCartesian coordinate system) and each position p_(k) in a selecteddirection from the z axis, and wherein, in accordance with theinvention, an arbitrarily selectable excerpt of a volume image of theexamination subject is obtained by two-dimensionally Fouriertransforming the aforementioned data set with respect to u_(i) and p_(k)to obtain a frequency space function, multiplying the frequency spacefunction by an interpolation function in one dimension of the frequencyspace and by a convolution core function in another dimension of thefrequency space to obtain an interpolated, convoluted product,multiplying the interpolated, convoluted product by a phase factor whichis dependent on a location of each point of the interpolated, convolutedproduct relative to a reconstruction volume in a locus space and therebyobtaining a file set of frequency space points, three-dimensionallygridding the final set of frequency space points onto points of athree-dimensional Cartesian grid with grid dimensions Δρ_(x), Δρ_(y) andΔρ_(z), freely selecting Δρ_(x), Δρ_(y) and Δρ_(z) to generate anarbitrarily selectable excerpt of the volume image, and bythree-dimensionally fast Fourier transforming the arbitrary excerpt intothe locus space. The excerpt transformed into the locus space is thendisplayed.

For generating an image in a plane x,y, which is arbitrarily oriented inspace, the above-described apparatus can be modified so that no griddingtakes place in the ρ_(z) direction, and Fourier transformation isinstead directly implemented for the position z=0. The gridding of thefinal set of frequency space points onto the Cartesian grid then takesplace only on the basis of a two-dimensional gridding, with thedimensions Δρ_(x) and Δρ_(y) being freely selectable. The Fouriertransformation into the locus space is then a two-dimensional fastFourier transformation.

DESCRIPTION OF THE DRAWINGS

FIG. 1 is a schematic illustration of an x-ray computed tomographyapparatus having EBT geometry for explaining the invention.

FIGS. 2-5 are geometrical illustrations for explaining the imagereconstruction in the x-ray computed tomography apparatus of FIG. 1 inaccordance with the principles of the present invention.

DESCRIPTION OF THE PREFERRED EMBODIMENTS

FIG. 1 shows an x-ray computed tomography apparatus having an annularx-ray source 2 surrounding a measuring field 1, a ring anode 3 beingarranged in the x-ray source 2. The ring anode 3 is scanned by anelectron beam 5 generated by an electron gun 6 for generating arotating, fan-shaped x-ray beam 4. The electron gun 6 is followed byfocusing coils 7. A vacuum in the x-ray source 2 is maintained by vacuumpumps 8. The electron beam 5 is deflected onto the ring anode 3 by amagnetic deflection coil 9 for generating the x-ray beam 4. The x-raysemerging after penetrating the examination subject in the measuringfield 1 are acquired by an annular radiation detector 10 that iscomposed of a row of detector elements 10a. The output signals of thedetector elements 10a are supplied to a computer 11 that calculates animage of the investigated slice of the examination subject therefrom andreproduces this image on a monitor 12. The measuring field 1 is a fieldin an opening 13 into which the examination subject is inserted. Thex-ray beam 4 rotates on the ring anode 3 due to deflection of theelectron beam 5 for irradiating the examination subject from differentdirections, rotating around the axis 4a.

A control unit 14 operates the deflection coil 9 such that the electronbeam 5 penetrates the x-ray source 2 concentrically relative to the ringanode 3 before the beginning of a scan procedure until it reaches aradiation trap 15 of, for example, lead at the closed end. Beforereaching the radiation trap 15, it is defocused by a defocusing means16. For conducting a scan procedure, the electron beam 5 is deflectedonto the ring anode 3 by the deflection coil 9 and scans the ring anode3 from its end 17 to its end 18. Five focus positions are shown inFIG. 1. In fact, there are substantially more discrete focus positions,for example 1,000. Preferably, however, the focus should be continuouslyshifted by a traveling field, so that the scanning is determined bymeans of the detector interrogation (sampling). The x-ray beam 4 thusrotates opposite the direction of the electron beam 5 and is shown inits final position in FIG. 1. The scan procedure is ended at that point.

A renewed set-up of the annularly guided electron beam 5 subsequentlyensues. A new scan procedure begins with the deflection thereof onto theend 17 of the ring anode 3.

It is also possible to scan the ring anode 3 with the electron beam 5 inthe clockwise direction, i.e. from its end 18 to its end 17.

The radiation detector 10 is arranged such with respect to the ringanode 3 such that the x-ray beam 4 can pass by it before the x-ray beam4 enters into the measuring field 1, and so that x-ray beam 4 isincident on the radiation detector 10 only after emerging from themeasuring field 1.

In the exemplary embodiment, the ring anode 3 is fashioned as a partialring, however, it can alternatively be fashioned as a full ring.

Geometry:

In EBT geometry, fan projections that are inclined by the angle φrelative to the x-y plane arise for discrete projection angles _(l) (φis referred to as the "gyratory angle"). If the plane in which the fanat the angle _(l) lies has a vertical spacing u_(i) from the coordinateorigin, the intersection of this plane with the z axis is z_(i) =u_(i)/cos φ.

It will be assumed that a parallel projection in the same plane arisesfor each such fan projection as a result of re-interpolation,characterized by _(l), φ and u_(i). This re-interpolation can beimplemented substantially more simply than the interpolation of paralleldata for φ=0 that initially seems desirable.

FIG. 2 and FIG. 3 illustrate the geometry, whereby FIG. 3 shows a viewonto the y-z plane.

The straight line g (n₁ direction) resides perpendicularly to the planeof the drawing, on the z axis. It proceeds through the point (0, 0,z_(i)) and describes the angle _(l) with the x-z plane.

The vector r_(g) for the line g is thus ##EQU1##

The projection plane belonging to the angle _(l) is placed through g,this projection plane being inclined by the angle φ relative to the x-yplane. The projection plane contains the straight line g⊥, defined bythe vector ##EQU2##

The vector n resides perpendicularly on the projection plane: ##EQU3##

The vectors n₁, n₂ and n define an orthogonal coordinate system.

It is expedient to characterize an attenuation value lying in theprojection plane established by _(l) and u_(i) in the following way:

1) By the distance u_(i) of the projection plane from the coordinateorigin:

    u.sub.i =r.sub.i ·n=z.sub.i cosφ

2) By the "projection angle" _(l)

3) By the distance p_(k) of the measured value in n₁ direction from thez axis.

One thus has a set of measured values f(u_(i), p_(k), _(l)).

The scan grid in the n₁ direction is a; the scan grid in the n directionis a.sub.⊥ ; N_(p) projections are registered per "revolution": ##EQU4##wherein a_(m) is the alignment.

The term du(_(l)) takes into consideration that the data are acquired as"spiral data". During the exposure, the measured subject moves with aconstant feed rate in the z direction relative to the rotating x-raybeam 4, so that the position in the n direction has changed by exactlya.sub.⊥ after one revolution (N_(p) projections) (and changed by a.sub.⊥/cos φ in the z direction).

The operating mode wherein the measured subject is stationary during acomplete revolution and is shifted by a.sub.⊥ /cos φ in the z directionafter every revolution is representable as a special case fordu(_(l))=0.

Possible definitions of a three-dimensional reference image for theFourier reconstruction:

When a three-dimensional image B₀ (r) is to be constructed from the lineintegrals f(u_(i), p_(k), _(l)), in order for this three-dimensionalimage to correctly quantitatively reproduce the subject attenuationvalues μ(r), the integrals f(u_(i), p_(k), _(l)) must linearlycontribute to B₀ (r): ##EQU5##

G_(lik) (r) must be the same for all r on straight lines parallel to theprojection line (u_(i), p_(k), _(l)), i.e. in the n₂ direction and cantherefore be dependent only on the distance of the point r fromproportion line (u_(i), p_(k), _(l)). This distance can in turn bedivided into the distance in the n direction d.sub.⊥ =r·n-u_(i) of thepoint r from the projection plane characterized by u_(i) and _(l) andinto the distance d=r·n₁ -p_(k) between the projection of the point intothe projection plane and the projection line.

Because the inclination angle φ in EBT geometry is small and because theconventional spiral scan must be contained in the representation as aspecial case for φ=0, the distance dependency is described by theproduct of two functions, each of which is respectively dependent on oneof the two distance components:

    G.sub.lik (r)=L.sub.lik (r·n.sub.1 -p.sub.k) h.sub.lik (r·n-u.sub.i)                                    (6)

when all projection values f(u_(i), P_(k), _(l)) are identicallytreated, ##EQU6## arises according to a standard scaling.

This image is considered as "reference image" for the Fourierreconstruction. The goal of a Fourier reconstruction method must be toreproduce B₀ (r) in the image region under consideration.

In a conventional spiral scan (φ=0), h(u) is the interpolation functionin the z direction. L₀ (p) is the normal convolution core whose Fouriertransform L₀ (ρ) is related to the modulation transfer function M_(A)(ρ) of the reconstruction in the following way: ##EQU7##

In the general case (φ≠0), ##EQU8## is valid for L₀ (ρ).

Derivation of a three-dimensional Fourier reconstruction method:reconstruction of the entire measurement volume:

Theoretical Description

Let the diameter of the measuring field in the x direction and in the ydirection be D_(M). Let the expanse (thickness) of the measuring fieldin the z direction be D_(z).

When

    L(p)=L.sub.0 (p) for |p|≦D.sub.M

    L(p)=0 for |p|>D.sub.M                   (10)

is set (exactly as in the two-dimensional case), the projections in then₁ direction convoluted with L(p) can be periodically continued withoutdegrading the image in the measuring field region, when

    w≦2D.sub.M

is valid for the period length w. The function h(u) will be of slightexpanse in the locus space anyway (expanse on the order of magnitude ofa slice width b; in the case of the spiral scan, h(u), for example, isthe linear interpolation between neighboring slices).

    v≦D.sub.z +b

The image B₁ (r) defined in the following way is identical to B₀ (r) inthe measuring field region: ##EQU9##

The three-dimensional Fourier transform ₁ (ρ) of this image reads:##EQU10##

The integral over (r·n₂) is a δ function and is derived as:

    ∫d(r·n.sub.2)exp(-2πi(ρ·n.sub.2)(r·n.sub.2))=δ(ρ·n.sub.2)                   (13)

Also valid is: ##EQU11##

When ρ=ρ·n₁ is set, then, because ##EQU12## the following is valid:##EQU13##

In the same way, one obtains ##EQU14## with Δρ.sub.⊥ =ρ·n and Δρ.sub.⊥=1/v.

Equations (13), (16) and (17) introduced into (12) leads to the result:##EQU15## with ρ=ρ·n₁ and ρ.sub.⊥ =ρ·n.

f(nΔρ.sub.⊥, mΔρ, _(l))is the two-dimensional, discrete Fouriertransform of f(u_(i), p_(k), _(l)) with respect to u_(i) and p_(k) :##EQU16## whereby Equation (4) was employed.

Equation (18) means that a continuous, three-dimensional image B₁ (r)was defined in the locus space whose three-dimensional Fourier transform₁ (ρ) in the frequency space exists only at discrete points.Therebetween, ₁ (ρ) has no values.

Values on a plane ρ·n₂ =0 in the frequency space belong to each"projection angle" _(l). This plane is inclined relative to the ρ_(x)-ρ_(z) plane by the "projection angle" _(l) and by the "gyroscopicangle" φ.

This is a generalization of a theorem referred to as the "central slicetheorem" in two-dimensions. In the present context, it can be stated asfollows:

The three-dimensional Fourier transform ₁ (ρ) of the image B₁ (r) on aplane in the frequency space that proceeds through the origin and isinclined relative to the ρ_(x) -ρ_(z) plane by the "projection angle"_(l) and by the "gyroscopic angle" φ is equal to the two-dimensionalFourier transform of the projections f(nΔρ.sub.⊥, mΔρ, _(l)) L(mΔρ)h(nΔρ.sub.⊥) registered for each angle _(l).

The values of the plane defined by and associated with the "projectionangle" _(l) are present in a Cartesian, discrete grid, having the griddimension Δρ in the n₁ direction and Δρ.sub.⊥ in the n direction.

In order to create the basis for a three-dimensional Fourierback-transformation of the spectrum into the locus space with FFTalgorithms, a new image B₂ (r) is defined, this arising from B₁ (r) bymultiplication with the step function T(r):

    T(r)=T.sub.1 (x)T.sub.1 (y)T.sub.2 (z)

    T.sub.1 (x)=1 for |x|≦D.sub.B /2

    T.sub.1 (x)=0 for |x|>D.sub.B /2

    T.sub.2 (z)=1 for |z|≦D.sub.Z /2

    T.sub.2 (z)=0 for |z|>D.sub.Z /2         (20)

D_(B) ·D_(B) is the central image excerpt of interest in the x-y plane,D_(Z) is the length of the measuring field in z direction. The image B₂(r) coincides with B₁ (r) in volume D_(B) ·D_(B) ·D_(Z) ; outside ofthis volume, it is zero. The periodic repetition of the image in thelocus space occurring due to the spectrum scanning in the Cartesiancoordinates given the three-dimensional FFT therefore does not lead tooverlap errors.

    B.sub.2 (r)=B.sub.1 (r)T.sub.1 (x), T.sub.1 (y), T.sub.2 (z) (21)

is set.

The following are then valid: ##EQU17##

With ρ=ρ·n₁, one can thus write: ##EQU18## The equality ##EQU19## wasthereby used (also see (15)).

With ρ.sub.⊥ =ρ·n, one likewise obtains: ##EQU20##

Equations (25) and (27) introduced into (22) results in: ##EQU21##

₁ (ρ), the three-dimensional Fourier transform of the image B₁ (r)unlimited in the locus space, is only defined at discrete points in thefrequency space (see equation (18)).

₂ (ρ), by contrast, the three-dimensional Fourier transform of the imageB₁ (r) multiplied by the step function T₁ (x) T₁ (y) T₂ (z) hascontinuous values in the frequency space. ₂ (ρ) arises by convolution ofthe discrete ₁ (ρ) with the one-dimensional Fourier transform T₁ (ρ_(x))T₁ (ρ_(y)) T₂ (ρ_(z)) of T₁ (x) T₁ (y) T₂ (z).

₂ (ρ) is obtained at the location (ρ_(x), ρ_(y), ρ_(z)) by calculatingthe distance of the point (ρ_(x), ρ_(y), ρ_(z)) from every point(nΔρ.sub.⊥, mΔρ, _(l)) of ₁ (ρ) in the ρ_(x) direction (i.e., ρ₁ =ρ_(x)-mΔρ cos _(l) -nΔρ.sub.⊥ sin _(l) sin φ) in the ρ_(y) direction (i.e.,ρ₂ =ρ_(y) -mΔρ sin _(l) +nΔρ.sub.⊥ cos _(l) sin φ), and in ρ_(z)direction (i.e., ρ₃ =ρ_(z) -nΔρ.sub.⊥ cos φ), the value of the point(nΔρ.sub.⊥, mΔρ, φ_(l)) is weighted with T₁ (ρ₁) T₁ (ρ₂)T₂ (ρ₃) and allsuch contributions are added.

The continuous spectrum ₁ (ρ) can then be scanned in the raster pointsαΔρ_(x), βΔρ_(y), γΔρ_(z) with

    Δρ.sub.x ≦1/D.sub.B

    Δρ.sub.y ≦1/D.sub.B

    Δρ.sub.z ≦1/D.sub.Z

which are transformed into the locus space with a three-dimensional FFTwithout aliasing errors occurring in the periodic repetition of theimage B₂ (r) following therefrom.

As in the two-dimensional case, the method can be simply expanded toarbitrary, non-central image excerpts D_(B) ·D_(B) in the x-y plane, asfollows. Let the desired reconstruction center lie at the location

    r.sub.z =(r.sub.z cos.sub.z, r.sub.z sin.sub.z, Z.sub.z)   (29)

A shift of the reconstruction center in the z direction is possible butnot actually meaningful because one could then just position the patientdifferently, or shorten the length of the scan region, in order to keepthe radiation stress as low as possible.

After the ideational shift of the image B₁ (r) by the vector -r_(z), sothat the reconstruction center again comes to lie on the coordinateorigin, one multiplies with the step function T₁ (x) T₁ (y) T₂ (z) inorder to obtain B₂ (r). Shift by -r_(z) in the frequency space meansmultiplication with a phase factor.

With

    r.sub.z ·n=r.sub.z (cos.sub.z cos.sub.l +sin.sub.z sin.sub.l)=r.sub.z cos(.sub.l -.sub.z)                    (30)

    r.sub.z ·n=r.sub.z (cos.sub.z sin .sub.l sinφ-sin .sub.z cos.sub.l sinφ)+z.sub.z cosφ=r.sub.z sinφsin(.sub.l -.sub.z)+z.sub.z cosφ                                 (31)

one obtains the following for three-dimensional Fourier reconstructionfrom EBT data: ##EQU22##

An arbitrary rotation of the illustrated image volume around the point(r_(z) cos _(z), r_(z) sin _(z), z_(z)) can also be realized withoutsignificant additional outlay, so that reformattings can be therebyreplaced up to a certain degree.

Simplifications for the practical realization:

As in the two-dimensional case, of course, the image B₁ (r) in the locusspace will not be limited with an ideal rectangular stop since theFourier transform thereof is

    T.sub.1 (ρ.sub.x)T.sub.1 (ρ.sub.y (T.sub.2 (ρ.sub.z)=D.sub.B.sup.2 D.sub.Z sinc(πρ.sub.x D.sub.B)sinc(πρ.sub.y D.sub.B)sinc(πρ.sub.z D.sub.z) (34)

so that every point of ₁ (ρ) contributes to every point of ₂ (ρ)(αΔρ_(x), βΔρ_(y), γΔρ_(z)). Instead, a volume D_(R) ·D_(R) ·D_(Z), isreconstructed that is larger than the desired image volume D_(B) ·D_(B)·D_(Z), and step functions T₁ and T₂ are selected such that T₁ decreasesor fades to an adequate extent along the path from D_(B) /2 to D_(R)-D_(B) /2 and T₂ decreases or fades to an adequate extent along the pathfrom D_(Z) /2 to D_(Z') -D_(Z) /2 and subsequently no longer upwardlyexceeds a smallest value ε_(min). Suitable functions, for example, arethe modified Van der Maas window, the Blackman window or a combinationof the two.

In the spiral mode, the data in the z direction arise in a densesequence for an EBT apparatus, i.e. the grid a.sub.⊥ is extremely small.When, for example, one sets a 3 mm slice and a patient feed of 3 mm persecond is undertaken in z the direction, a revolution will lastapproximately 50 ms, with 20 revolutions per second, and thus, ##EQU23##given a gyroscopic angle of 0.5°.

Approximately 400 slices are obtained in the z direction for D_(z) =60mm, so that the two-dimensional Fourier transformation of theprojections f(u_(i), p_(k), _(l)) in the n₁ direction (p_(k)) would, asbefore, have to be of the dimension 2048 or 4096 dependent on the numberof detector elements, also of the dimension 512 in the directionn(u_(i)) (impractically large).

Since the slice thickness b (for example 3 mm), however, issignificantly larger than the spacing of neighboring slices (a.sub.⊥=0.15 mm), a number of projections that have arisen given the sameprojection angle _(l) and follow one another in the n direction (u_(i)with ascending index i) can be combined, so that an effective a.sub.⊥ ofapproximately half the slice thickness b arises and thus only 64supporting points, for example, for the Fourier transmission of theprojection f(u_(i), p_(k), _(l)) in the n direction.

The unsharpening of the image in z direction that is unavoidable in thiscombination can be compensated by including a steepening part inh(nΔρ.sub.⊥).

Quasi-two-Dimensional Fourier Reconstruction of Individual Slices:

The three-dimensional Fourier reconstruction of the entire measurementvolume makes high demands of storage space and calculating speed.

In the example that has been mentioned (3 mm slice, length of themeasurement field in the z direction D_(Z) =60 mm), the projectionsf(u_(i), p_(k), _(l)) must first be transformed with two-dimensionalFFTs of the length 2048 ·64 into the frequency space for everyprojection angle _(l) (given 1024 detector elements). When, for example,1000 projections arise per revolution, 1000 of these two-dimensionalFFTs then must be implemented. The multiplication by L(mΔρ) andh(nΔρ.sub.⊥) as well as by the phase factor subsequently ensues in thefrequency space. _(z),1 (ρ) is thus defined. When a two-dimensionalimage having a 512·512 matrix is presented and when one wishes theimages in approximately the spacing of half the slice thickness, i.e.approximately 40-50 images for D_(Z) =60 mm, then--due to the propertiesof the step function T₁ (x) T₁ (y) T₂ (z)--the three-dimensional Fourierback-transformation into the locus space must be of the dimension1024·1024·128, i.e. _(z),2 (ρ_(x), ρ_(y), ρ_(z)) is required at just asmany supporting points. When T₁ and T₂ are suitably selected, _(z),1 (ρ)contributes to approximately 4·4·4 points of _(z),1 (ρ_(x), ρ_(y),ρ_(z)).

It is also a disadvantage that one cannot begin with the reconstructionuntil all data have been registered, i.e. until after the entire scan.

For these reasons, it may be desirable to reconstruct successivetwo-dimensional images as before (for instance, in the spacing of half aslice thickness). Only a relatively small data set is then required forthe reconstruction of the first image.

A quasi-two-dimensional Fourier construction method for EBT data shallbe set forth below.

Is seen in the y-z plane, FIG. 4 shows the desired slice at z₀ as wellas the region Δu in the n direction from which data contribute to theimage at z₀. When ##EQU24## is introduced, then u_(i) must be taken intoconsideration for

    l.sub.0 -l≦i≦l.sub.0 +l                      (36)

Analogous to equation (11), an image B₁,l.sbsb.0 (r) is nowdefined--although two-dimensionally--at the location r=(x, y, z₀),whereby the projections f(u_(i), p_(k), _(l)) in the n₁ directionconvoluted with L(p) and h(u) are repeated as in (11) with periodw=2D_(M), but with the period v'=Δu+B in the ndirection (b is theexpanse of h(u)): ##EQU25## Because

    r·n.sub.1 =xcos.sub.l +ysin.sub.l                 (38)

    r·n=xsin.sub.l sinφ-ycos.sub.l sinφ=z.sub.0 cosφ(39)

the two-dimensional Fourier transform of this image is: ##EQU26##

Following therefrom with equations (25) and (27): ##EQU27## withΔρ'.sub.⊥ =1/v' and--as previously--Δρ=1/w.

When the substitution

    j=i-l.sub.0                                                (42)

is implemented then based on Equation (4)

    u.sub.i =U.sub.j+l.sub.0 =ja.sub.⊥ +l.sub.0 a.sub.195 +du(.sub.l)=u.sub.j +l.sub.0 a.sub.⊥                 (43)

with

    z.sub.0 cosφ-l.sub.0 a.sub.⊥ =dz.sub.0 cosφ   (44)

the following is obtained for B₁,l.sbsb.0 (ρ_(x), ρ_(y), Z_(o)):##EQU28##

The term f_(l) ₀ (nΔρ'.sub.⊥, mΔρ, _(l)) is thereby the two-dimensionalFourier transform of the projection f(u_(j+l) ₀, p_(k), _(l)): ##EQU29##

B₁,l.sbsb.0 (ρ_(x), ρ_(y), Z_(o)) in the two-dimensional ρ_(x) -ρ_(y)frequency space is also defined only at discrete points, namely at thelocations δ(ρ_(x) -mΔρcos_(l) -nΔρ'.sub.⊥ sin_(l) sinφ) and δ(ρ_(y)-mΔρsin_(l) +nΔρ'.sub.⊥ cos_(l) sinφ).

This becomes δ(ρ_(x) -mΔρcos_(l))δ(ρ_(y) -mΔρsin_(l)) for φ=0(projections perpendicularly on the z axis). The points--as in theconventional, two-dimensional case--then lie on a polar grid in theρ_(x) -ρ_(y) plane.

In order to make the two-dimensional image B₁,l.sbsb.0 (x, y, z₀) (z₀ isonly a parameter, no longer a variable) useable for the two-dimensionalFourier reconstruction, it is multiplied by the step function T₁ (x) T₁(y) (see (20) for definition) and B₂,l₀ (X, y, z₀), the lattercoinciding with B₁,l.sbsb.0 (x, y, z₀), in an initially central imageexcerpt D_(B) ·D_(B) :

    B.sub.2,l.sub.0 (x,y,z.sub.0)=B.sub.1,l.sbsb.0 (x,y,z.sub.0)T.sub.1 (x)T.sub.1 (y)                                            (47)

The two-dimensional Fourier transform of this image is calculated as:##EQU30##

B₂,l₀ (ρ_(x), ρ_(y), Z_(o)) is continuous and--as required fortwo-dimensional FFT--can be scanned in the Cartesian scan pointsαΔρ_(x),βΔρ_(y), with

    Δρ.sub.x ≦1/D.sub.B

    Δρ.sub.y ≦1D.sub.B                        (49)

As in the three-dimensional case, the expansion to non-central imageexcerpts D_(B) ·D_(B) in the x-y plane is simple. With r_(z) =(r_(z)cos_(z), r_(z) sin_(z), 0) for the position of the reconstructioncenter, one obtains: ##EQU31##

The following estimate of the outlay for the reconstruction of anindividual slice at z₀ is of interest.

As in the three-dimensional reconstruction, the projections that arosegiven the same projection angle _(l) can be combined for a number ofsuccessive u_(i), so that an effective grid a.sub.⊥ having approximatelyhalf a slice thickness arises. l≈2 is then valid, so that theprojections f(u_(j+l) ₀, p_(k), _(l)) (given 1024 detector elements) areto be transformed into the frequency space for every projection angle_(l) having a two-dimensional FFT of the length 2048·4. Aftermultiplication by L(mΔρ)h(nΔρ'.sub.⊥) and the corresponding phasefactor, each of the supporting points contributes to approximately 4·4supporting points of the Cartesian grid for the two-dimensional Fourierback-transformation that, as usual, ensues with 1024·1024 values.

A not unsubstantial difference compared to three-dimensionalreconstruction lies in the switch to the Cartesian grid: every pointtherein contributes to 4·4·4 supporting points of the three-dimensionalCartesian grid, a significant advantage.

The total outlay for the production of an individual image should,according to these preliminary estimates, lie at about 3-4 times theoutlay for the production of an individual image from conventional,two-dimensional parallel data.

As in the three-dimensional reconstruction, it is easily possible torotate the two-dimensional discrete slice in space on the basis of acoordinate transformation.

Derivation of L₀ (ρ):

The relationship ##EQU32## is explained in this section (see Equation(9)).

In continuous notation, the reference image (from Equation (7)) reads:##EQU33##

When data are obtained from a uniform circular cylinder having adiameter D and attenuation μ that has an infinite expanse in the zdirection the examination as subject, then ##EQU34## is valid, and thus:##EQU35##

With r·n₁ =x cos +y sin , the two-dimensional Fourier transform of theindividual slice calculated at an arbitrary location z₀ is ##EQU36##Because

    ∫ dx exp (-2πix(ρ.sub.x -ρcos))=δ(ρ.sub.x -ρcos)=δ(ρ.sub.x -ρ.sub.x)              (55)

    ∫ dy exp (-2πiy(ρ.sub.y -ρsin ))=δ(ρ.sub.y -ρsin)=δ(ρ.sub.y -ρ.sub.y)              (56)

and

    |ρ|dρd=dρ'.sub.x dρ'y    (57)

one can thus write: ##EQU37##

With M_(A) (ρ) as a modulation transfer function and O(ρ) as thetwo-dimensional Fourier transform into the ρ_(x) -ρ_(y) plane of thecircular cylinder with infinite expanse in the z direction, thefollowing is simultaneously valid: ##EQU38## Following therefrom,##EQU39##

Coordinate Transformation and Derivation of the Reconstruction Equationsfor Arbitrarily Rotated, Individual Slices:

FIG. 5 illustrates the first step of the coordinate transformation. Thestarting point is the coordinate system x, y, z. The new coordinatesystem x', y', z' is shifted (-x_(z), -y_(z), -z_(z)) and is rotated bythe angle δ. The x' axis is the rotational axis. Thus

    x'=x+x.sub.z                                               (61)

    y'=(Y+y.sub.z)cos δ+(z+z.sub.z)sin δ           (62)

    z'=-(y+y.sub.z)sin δ+(z+z.sub.z cos δ          (63)

The coordinate system x', y', z' is subsequently rotated by the angle γ.The y' axis is the rotational axis. The coordinate system x", y", z" isobtained with

    x"=x'cosγ+z'sinγ                               (64)

    z"=-x'sinγ+z'cosγ                              (65)

    y"=y'                                                      (66)

The overall transformations are:

    x=x"cosγ-z"sinγ-x.sub.z                        (67)

    y=y"cosδ-x"sinγsinδ-z"cosγsinδ-y.sub.z (68)

    z=x"sinγcosδ+z"cosγcosδ+y"sinδ-z.sub.z (69)

The individual slice in the coordinate system x", y", z" is observed atthe location z"=0 (otherwise, x_(z), y_(z), z_(z) could have beendifferently selected). Then

    x=x"cosγ-x.sub.z                                     (70)

    y=y"cosδ-x"sinγsinδ-y.sub.z              (71)

    z=x"sinγcosδ+y"sinδ-z.sub.z              (72)

The starting point for the image description is equation (37): ##EQU40##

Following therefrom with Equations (10), (11), (12): ##EQU41##

The two-dimensional Fourier transform with respect to x" and y" at thelocation z"=0 is ##EQU42## with the substitution

    j=i-l.sub.0                                                (76)

    u.sub.i =u.sub.j +l.sub.0 =ja.sub.⊥ +l.sub.0 a.sub.⊥ +du(.sub.i)=u.sub.j +l.sub.0 a.sub.⊥                 (77)

    -z.sub.z cosφ-l.sub.0 a.sub.⊥ =dz.sub.0 cosφ  (78)

this becomes ##EQU43##

f_(l).sbsb.0 (nΔρ.sub.⊥, mΔρ_(l)) is the two-dimensional Fouriertransform of f(u_(j+l).sbsb.0, p_(k), _(l)): ##EQU44##

The image B_(1D),l.sbsb.0 (x", y", 0) is multiplied by the step functionT₁ (x"), T₁ (y") in the new coordinate system x", y", z" and the imageB_(2D),l.sbsb.0 (x", y", 0) is obtained with the Fourier transform##EQU45##

The sole difference compared to equation (50) is that the weightingfunctions T₁ (ρ_(x")) T₁ (ρ_(y")) are to be calculated at otherlocations because of the rotated coordinate system. This, however, doesnot involve added outlay because γ and δ are constants. Added outlaydoes arise, however, because individual scans must be utilized forconstructing a slice under certain circumstances, i.e. l becomes larger.

When δ=0 and γ=0 is set, equation (50) is obtained.

Although modifications and changes may be suggested by those skilled inthe art, it is the intention of the inventor to embody within the patentwarranted hereon all changes and modifications as reasonably andproperly come within the scope of his contribution to the art.

I claim as my invention:
 1. An x-ray computed tomography apparatuscomprising:an annular x-ray source containing a ring anode surrounding ameasuring field and means for scanning said ring electrode with anelectron beam for producing an x-ray beam rotating around said measuringfield through successive projection angles _(l) in a plane inclined atan angle φ relative to an x-y plane of a Cartesian coordinate systemhaving an origin, said x-y plane being disposed a distance z_(i) fromsaid origin; means for operating said annular x-ray source forconducting a scan in parallel beam geometry of an examination subjectdisposed in said measuring field by irradiating said examination subjectwith said x-ray beam once from each of said successive projection angles_(l) ; radiation detector means for detecting said x-ray beam duringsaid scan after passing through said examination subject from each ofsaid projection angles _(l) to obtain a set of measured values f(u_(i),p_(k), φ_(l)) for each scan for each projection angle _(l) and eachposition u_(i) =z_(i) /cosφ and each position p_(k) in a selecteddirection from the z-axis of said Cartesian coordinate system; means forgenerating a representation of a volume image of said examinationsubject from said set of measured data by two-dimensionally Fouriertransforming said set of measured data with respect to u_(i) and p_(k)to obtain a frequency space function, multiplying said frequency spacefunction by an interpolation function in one dimension of said frequencyspace and by a convolution core function in another dimension of saidfrequency space to obtain an interpolated, convoluted product,multiplying said interpolated, convoluted product by a phase factorwhich is dependent on a location of each point of said interpolated,convoluted product relative to a reconstruction volume in a locus spaceto obtain a final set of frequency space points, three-dimensionallygridding said final set of frequency space points onto points of athree-dimensional Cartesian grid with grid dimensions Δρ_(k), Δρ_(y) andΔρ_(z), freely selecting Δρ_(x), Δρ_(y) and Δρ_(z) to generate anarbitrarily selected excerpt of said representation of said volumeimage, and by three-dimensionally fast Fourier transforming saidarbitrarily selected excerpt into said locus space; and means fordisplaying the arbitrarily selected excerpt transformed into said locusspace.
 2. An x-ray computed tomography apparatus comprising:an annularx-ray source containing a ring anode surrounding a measuring field andmeans for scanning said ring electrode with an electron beam forproducing an x-ray beam rotating around said measuring field throughsuccessive projection angles _(l) in a plane inclined at an angle φrelative to an x-y plane of a Cartesian coordinate system having anorigin, said x-y plane being disposed a distance z_(i) from said origin;means for operating said annular x-ray source for conducting a scan inparallel beam geometry of an examination subject disposed in saidmeasuring field by irradiating said examination subject with said x-raybeam once from each of said successive projection angles _(l) ;radiation detector means for detecting said x-ray beam during said scanafter passing through said examination subject from each of saidprojection angles _(l) to obtain a set of measured values f(u_(i),p_(k),φ_(l)) for each scan for each projection angle _(l) and each positionu_(i) =z_(i) /cos φ and each position p_(k) in a selected direction fromthe z-axis of said Cartesian coordinate system; means for generating arepresentation of a planar image of said examination subject from saidset of measured data by two-dimensionally Fourier transforming said setof measured data with respect to u_(i) and p_(k) to obtain a frequencyspace function, multiplying said frequency space function by aninterpolation function in one dimension of said frequency space and by aconvolution core function in another dimension of said frequency spaceto obtain an interpolated, convoluted product, multiplying saidinterpolated, convoluted product by a phase factor which is dependent ona location of each point of said interpolated, convoluted productrelative to a reconstruction volume in a locus space to obtain a finalset of frequency space points, two-dimensionally gridding said final setof frequency space points onto points of a two-dimensional Cartesiangrid with grid dimensions Δρ_(x), and Δρ_(y), freely selecting Δρ_(x)and Δρ_(y) to generate an arbitrarily selected excerpt of said planarimage, two-dimensionally fast Fourier transforming said arbitrarilyselected excerpt into said locus space and directly fast Fouriertransforming said final set of frequency space points for z=0 into saidlocus space; and means for displaying said arbitrarily selected excerpttransformed into said locus space.